Mince 1 onion finely. Heat 1 Tbsp extra virgin olive oil in a large pan on medium-high heat and sauté the onion until translucent. Turn off the heat. Divide the sautéed onion to use for the gravy and the hamburger patties.

Cut 6 button mushrooms into thin slices. Heat a small saucepan on medium heat. Then, add ½ Tbsp extra virgin olive oil and the mushrooms and sauté until soft. Add half of the sautéed onion and stir together. Add 1½ cups beef stock/broth, 1 tsp Worcestershire sauce, and 1 tsp soy sauce. Season with Diamond Crystal kosher salt to taste and ⅛ tsp freshly ground black pepper. Make a slurry with 1 Tbsp potato starch or cornstarch and 1 Tbsp water in a small bowl and whisk well to combine. Pour it into the gravy and thoroughly mix together. Simmer until the gravy thickens, then turn off the heat and set aside.

Add ½ lb ground beef, ¼ lb ground pork, and the other half of the sautéed onion to a large bowl. Add ⅓ cup panko, 1 large egg, 2 Tbsp milk, 1 tsp Worcestershire sauce, ½ tsp nutmeg, 1 tsp Diamond Crystal kosher salt, and ⅛ tsp freshly ground black pepper. Knead together well using your hand until the mixture becomes sticky and pale. Then, divide into 4 portions for large patties or 6 portions for medium patties. Make an oval-shaped patty with a slightly rounded top. Cover with plastic wrap and rest them in the refrigerator for at least 30 minutes.

Heat a large pan over medium heat, then add ½ Tbsp extra virgin olive oil. Place the patties gently in the pan. Indent the center of each patty with 2 fingers. Fry them for about 5 minutes or until browned on the bottom. Flip over, then cover with a lid and fry until thoroughly cooked inside, about 5 minutes. At this time, reheat the gravy and make the fried eggs with 4 large eggs and ½ Tbsp extra virgin olive oil.

Divide 4 servings cooked Japanese short-grain rice on individual plates. Transfer the patties to the plates, pour the gravy on top, and top each with a fried egg. Chop 1 sprig parsley and sprinkle on top for garnish. Serve immediately.